Before I extol the virtues of the train system in Japan, there is one thing that you have to understand first. It is expensive compared to transit services in many North American cities. As far as I know, it's not subsidised by government, so you have to pay the full cost. The main rail system is owned by Japan Rail, but they are operated by individual companies -- each with a monopoly in their own area. Additionally, there are "private" train companies that both own the lines and run the trains. If you live in a very rural area (and it wasn't screwed over by the bus companies), you will almost certainly be using these private lines. If you want to go to a big city, it usually means that you have to transfer. Luckily the payment options are almost completely harmonised -- at least on the regional level. You can get pre-paid cards that work for all the companies in your region (with a very few exceptions).
I live in Shizuoka prefecture. To take the bus a distance of about 30 km it costs about $10 US (also not subsidised). To take the train, it's a bit cheaper (about $7). If you take intercity busses, the relationship switches. A 200 km bus trip costs about $30, while the train will cost about $50. However, the intercity busses only travel about 4 times a day, while the trains come every 15 minutes.
The main difference between the train and the bus is convenience and speed. The trains are quite fast. I often travel to Hamamatsu city from Kanaya city by train. Due to circumstances, my wife goes by car. I beat her every single time -- sometimes by hours. By car, there are lots of unexpected events: accidents, traffic jams, etc. On the train, there are very few -- really just suicide attempts and I haven't seen one in our area for years.
In Japan the trains are on time. Very, very, very occasionally the train will be 1 minute late. Once a year it will be 2 or more minutes late. Seriously, if the train is not on the platform 30 seconds before the scheduled departure time, people are looking at their watches in disbelief.
Part of this is because the trains are set up so that freight is generally able to pass and be passed. On private, rural lines, there is never any freight (though the trains often carry mail). Also the trains are all in good shape. I've literally never been on a train in Japan that broke down. When I lived just outside of London, the trains would break down every week at least. One of the other main points is that theft in Japan is virtually unheard of. When I lived in the UK, I heard that one of the reasons the signalling constantly broke down is because people would steal the copper cables for scrap. Culturally, this just doesn't happen in Japan, so it's a consideration you have to keep in mind if you want to emulate Japan's success with trains.
While my wife has a car, I don't like using it. If we go anywhere, it's usually by bus and then by train. It's only if we need a car at the other end that we use the car (for example, when we go to Hamamatsu, my wife takes the car so that she can take her mother shopping, etc). Because there is no train where I live, I end up always taking the bus.
Even though Japan is a country where punctuality is almost taken for granted, busses are not punctual. They suffer from the same problems that happen everywhere -- busses are really hard to predict. Sometimes the road is clear, and sometimes it is not. The busses usually end up being virtually randomly timed by the time they end up in the middle of their route. I almost always miss my connection at the train station. Because the bus comes every hour or so, it means I usually have to plan very carefully to ensure that I arrive on time. Often I end up arriving an hour early just to make sure I get there on time.
Before I got married, I went out socially a lot more. I discovered that even though the train station is 20 km away, it was faster (and more predictable) for me to ride my bike there, take a bath at the nearby hot springs and then go to the city by train than it was to simply take the bus there directly. This is where I got my love of cycling (and hot springs!).
As I alluded to earlier, a lot of the reason that Japanese trains work is due to Japanese culture. For example, companies that get a monopoly will rip you off, but only by so much. They truly have a sense of social responsibility. The UK moved from a national rail system to something similar to what Japan has and from what I can tell, it completely broke their rail system (I'll probably get some argument about that). However, from my perspective, it's pretty clear that rail is simply easier to make work well than busses. It doesn't necessarily follow that any area building a rail system will make one that works well (I come from Canada and though I love trains I am never going to give any more money to Via. Ever.)